We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Langerhans. Long term goals are to understand how the membrane dynamics interact with intracellular events to regulate secretion. We also compare, contrast, and generalize to other secretory cells and neurons, including GnRH-secreting hypothalamic neurons, pituitary somatotrophs, and fast neurotransmitter secretion at nerve terminals. Our primary tool is the numerical solution of ordinary and partial differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get analytical results in special cases. We study both detailed biophysical models and simplified models which are more amenable to analysis. Such an approach aids the isolation of the essential or minimal mechanisms underlying phenomena, the search for general principles, and the application of concepts and analogies from other fields. Another role for our group is to mediate between the mathematical and biological disciplines. This includes disseminating the insights of mathematical work to biologists in accessible language and alerting mathematicians and other theoreticians to new and challenging problems arising from biological issues. Recent work on this project includes: 1. (Role of the Endoplasmic Reticulum in Shaping Calcium Oscillations) We have shown that adding a simple ER with only linear uptake and release mechanisms is sufficient to account for most features of cytosolic Ca2+ kinetics, provided the ER is much slower than cytosolic Ca2+, but not too slow. It must be able to fill and empty substantially during a burst cycle (tens to hundreds of seconds) in order to impart its slow kinetics to cytosolic Ca2+. Inclusion of ER dynamics is sufficient to account for the increase of burst frequency in the presence of the insulin-secretion potentiator acetylcholine. Inclusion of nucleotide ratio dynamics permits in addition simulation of the triphasic transient response of islets to a step of glucose (latency, first phase spiking, and steady-state oscillation). See Bertram and Sherman (2004). Not all additional mechanisms are helpful, however. We have found that including active calcium-induced calcium release (CICR) results in an ER that does not fill and empty in response to cytosolic calcium oscillations and fails to account for the increase in the amplitude of cytosolic calcium transients when ER uptake is blocked. A paper is in press. 2. (Electrical Coupling and Emergent Oscillations in Pancreatic Islets) We have extended our work on how the heterogeneous properties of islet beta-cells contributes to the collective behavior of intact islets. Using the phantom bursting model mentioned above (Bertram and Sherman, 2004), we have shown that coupling fast and slow cells can produce the intermediate period electrical oscillations typically seen in islets. This is not very surprising, but we found further that a bimodal distribution of single-cell periods could be generated with a unimodal distribution of channel conductances. It is also possible to construct islets consisting of only fast cells or only slow cells that exhibit intermediate period oscillations when coupled. See Zimliki et al (2004). 3. (Combined Electrical and Metabolic Oscillations in Pancreatic Islets) Although electrical osscillations in pancreatic islets are important for understanding many phenomena, their properties are at variance with observations of pulsatile insulin secretion in vivo. We have proposed that this can be explained by the modulation of the electrical oscillations by metabolic (glycolytic) oscillations. In particular, this combination can account for the observations of compound oscillations (bursts of bursts) that have been observed in membrane potential, cytosolic calcium, and metabolic variables such as intra-islet oxygen and glucose and mitochondrial membrane potential. We suggest that the glycolytic oscillations maintain optimal timing to coordinate insulin secretion and insulin action while the electrical oscillations control the quantity of insulin secreted in each pulse. See Bertram et al (2004).